Part IVGame‐Theoretic Finance

We now look at how game‐theoretic probability in continuous time can help us understand the emergence of Brownian motion, the foundations of the Itô calculus, and other aspects of continuous‐time martingales. These ideas are widely used in physics, in other branches of science, and especially in finance theory. We emphasize finance theory, because it has inspired most of the work on continuous‐time martingales in recent decades and because its game‐theoretic nature is already so evident.

We develop game‐theoretic finance theory at the same abstract and idealized level used in measure‐theoretic finance theory. For simplicity, we assume that financial securities' price paths are continuous, even though most of the results we report can be generalized to accommodate some discontinuities. More importantly, we assume that securities do not pay dividends and that their shares can be frequently bought, sold, and shorted in arbitrary fractional amounts with no transaction costs. The implications of our results for any actual financial market must be tempered by taking into account the degree to which it departs from this extremely idealized picture.

Measure‐theoretic finance theory assumes that a financial security's successive prices are described by a stochastic process in continuous time. A stochastic process being a family of random variables in a probability space, this assumption seems to involve a hugely complex probability measure. But a celebrated ...

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